Peaceman-Rachford (alternating directions) scheme.

n this section we present a general way to construct a finite difference
approximation for a solution of the heat equation via a procedure with linear
dependency of the amount of computation on the size of the lattice.

We are considering the following boundary problem for the heat
equation:

We introduce the uniform
lattices
the lattice function
and notation
.
The introduced above notations
and
refer to the
-variable.
Consider the
scheme

(Alternating directions1)

(Alternating directions2)

in all internal points of the lattice
.
The
refers to the operator
acting in the
index. The initial conditions
are
and the boundary conditions
are